Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3321,2,Mod(1,3321)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3321, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3321.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3321 = 3^{4} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3321.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(26.5183185113\) |
| Analytic rank: | \(1\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 20 x^{14} + 158 x^{12} - 2 x^{11} - 629 x^{10} + 25 x^{9} + 1329 x^{8} - 116 x^{7} - 1433 x^{6} + \cdots + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 369) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} - 20 x^{14} + 158 x^{12} - 2 x^{11} - 629 x^{10} + 25 x^{9} + 1329 x^{8} - 116 x^{7} - 1433 x^{6} + \cdots + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3 \nu^{15} - 7 \nu^{14} + 35 \nu^{13} + 99 \nu^{12} - 74 \nu^{11} - 444 \nu^{10} - 423 \nu^{9} + \cdots - 24 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{15} - 7 \nu^{14} + 48 \nu^{13} + 125 \nu^{12} - 269 \nu^{11} - 847 \nu^{10} + 591 \nu^{9} + \cdots + 80 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8 \nu^{15} - 3 \nu^{14} - 167 \nu^{13} + 35 \nu^{12} + 1363 \nu^{11} - 90 \nu^{10} - 5476 \nu^{9} + \cdots + 25 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5 \nu^{15} - 3 \nu^{14} + 106 \nu^{13} + 74 \nu^{12} - 873 \nu^{11} - 675 \nu^{10} + 3507 \nu^{9} + \cdots + 64 ) / 13 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 17 \nu^{15} + 18 \nu^{14} - 311 \nu^{13} - 314 \nu^{12} + 2196 \nu^{11} + 2048 \nu^{10} - 7626 \nu^{9} + \cdots + 6 ) / 13 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 17 \nu^{15} - 18 \nu^{14} + 324 \nu^{13} + 340 \nu^{12} - 2391 \nu^{11} - 2451 \nu^{10} + 8627 \nu^{9} + \cdots + 72 ) / 13 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 22 \nu^{15} + 8 \nu^{14} - 430 \nu^{13} - 167 \nu^{12} + 3277 \nu^{11} + 1319 \nu^{10} - 12316 \nu^{9} + \cdots - 19 ) / 13 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 21 \nu^{15} - 23 \nu^{14} + 401 \nu^{13} + 420 \nu^{12} - 2975 \nu^{11} - 2900 \nu^{10} + 10871 \nu^{9} + \cdots + 27 ) / 13 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 25 \nu^{15} - 15 \nu^{14} + 478 \nu^{13} + 292 \nu^{12} - 3546 \nu^{11} - 2166 \nu^{10} + 12907 \nu^{9} + \cdots + 47 ) / 13 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 24 \nu^{15} + 17 \nu^{14} - 462 \nu^{13} - 324 \nu^{12} + 3465 \nu^{11} + 2356 \nu^{10} - 12840 \nu^{9} + \cdots - 120 ) / 13 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 43 \nu^{15} + 31 \nu^{14} - 831 \nu^{13} - 600 \nu^{12} + 6239 \nu^{11} + 4427 \nu^{10} - 22992 \nu^{9} + \cdots - 98 ) / 13 \)
|
| \(\beta_{15}\) | \(=\) |
\( 3 \nu^{15} + 3 \nu^{14} - 58 \nu^{13} - 57 \nu^{12} + 436 \nu^{11} + 414 \nu^{10} - 1612 \nu^{9} + \cdots - 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{7} + \beta_{6} + 6\beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{15} - 2 \beta_{14} - 2 \beta_{12} - \beta_{11} - \beta_{8} + 3 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{15} - 10 \beta_{14} - 9 \beta_{13} - 10 \beta_{12} - \beta_{11} + \beta_{9} - 2 \beta_{8} + \cdots + 63 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12 \beta_{15} - 22 \beta_{14} - 2 \beta_{13} - 21 \beta_{12} - 10 \beta_{11} + \beta_{10} - 10 \beta_{8} + \cdots + 10 \)
|
| \(\nu^{8}\) | \(=\) |
\( 75 \beta_{15} - 76 \beta_{14} - 64 \beta_{13} - 77 \beta_{12} - 12 \beta_{11} + 11 \beta_{9} + \cdots + 329 \)
|
| \(\nu^{9}\) | \(=\) |
\( 104 \beta_{15} - 179 \beta_{14} - 28 \beta_{13} - 165 \beta_{12} - 75 \beta_{11} + 13 \beta_{10} + \cdots + 81 \)
|
| \(\nu^{10}\) | \(=\) |
\( 511 \beta_{15} - 525 \beta_{14} - 421 \beta_{13} - 540 \beta_{12} - 103 \beta_{11} + 90 \beta_{9} + \cdots + 1812 \)
|
| \(\nu^{11}\) | \(=\) |
\( 792 \beta_{15} - 1303 \beta_{14} - 267 \beta_{13} - 1170 \beta_{12} - 510 \beta_{11} + 117 \beta_{10} + \cdots + 628 \)
|
| \(\nu^{12}\) | \(=\) |
\( 3344 \beta_{15} - 3478 \beta_{14} - 2682 \beta_{13} - 3625 \beta_{12} - 778 \beta_{11} + \beta_{10} + \cdots + 10359 \)
|
| \(\nu^{13}\) | \(=\) |
\( 5649 \beta_{15} - 8990 \beta_{14} - 2176 \beta_{13} - 7922 \beta_{12} - 3327 \beta_{11} + 905 \beta_{10} + \cdots + 4767 \)
|
| \(\nu^{14}\) | \(=\) |
\( 21484 \beta_{15} - 22593 \beta_{14} - 16852 \beta_{13} - 23786 \beta_{12} - 5520 \beta_{11} + \cdots + 60772 \)
|
| \(\nu^{15}\) | \(=\) |
\( 38829 \beta_{15} - 60232 \beta_{14} - 16352 \beta_{13} - 52413 \beta_{12} - 21300 \beta_{11} + \cdots + 35420 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.41518 | 0 | 3.83310 | 1.58894 | 0 | −3.32994 | −4.42726 | 0 | −3.83758 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.24183 | 0 | 3.02582 | 1.34218 | 0 | 0.228831 | −2.29971 | 0 | −3.00895 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.76241 | 0 | 1.10611 | −4.06339 | 0 | −5.00725 | 1.57541 | 0 | 7.16138 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.64102 | 0 | 0.692958 | −0.343705 | 0 | −0.00363542 | 2.14489 | 0 | 0.564028 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.55821 | 0 | 0.428025 | −0.552119 | 0 | 3.82874 | 2.44947 | 0 | 0.860319 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.948889 | 0 | −1.09961 | 2.82118 | 0 | 0.434589 | 2.94119 | 0 | −2.67698 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.357942 | 0 | −1.87188 | −2.56278 | 0 | −1.28872 | 1.38591 | 0 | 0.917326 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.153310 | 0 | −1.97650 | 3.79315 | 0 | −2.96558 | 0.609637 | 0 | −0.581529 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.223707 | 0 | −1.94995 | −0.189367 | 0 | 2.75447 | −0.883634 | 0 | −0.0423628 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.746256 | 0 | −1.44310 | 2.36325 | 0 | −0.386229 | −2.56944 | 0 | 1.76359 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.790369 | 0 | −1.37532 | 0.138797 | 0 | −4.64945 | −2.66775 | 0 | 0.109701 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.969698 | 0 | −1.05969 | −1.78868 | 0 | 1.82212 | −2.96697 | 0 | −1.73448 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.90913 | 0 | 1.64476 | −0.225395 | 0 | −1.26124 | −0.678190 | 0 | −0.430307 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.93363 | 0 | 1.73893 | −3.73785 | 0 | 0.466713 | −0.504807 | 0 | −7.22763 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.97529 | 0 | 1.90179 | 1.60012 | 0 | −0.734638 | −0.193998 | 0 | 3.16071 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.53072 | 0 | 4.40455 | −1.18434 | 0 | −2.90879 | 6.08526 | 0 | −2.99723 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(41\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3321.2.a.i | 16 | |
| 3.b | odd | 2 | 1 | 3321.2.a.j | 16 | ||
| 9.c | even | 3 | 2 | 369.2.e.a | ✓ | 32 | |
| 9.d | odd | 6 | 2 | 1107.2.e.a | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 369.2.e.a | ✓ | 32 | 9.c | even | 3 | 2 | |
| 1107.2.e.a | 32 | 9.d | odd | 6 | 2 | ||
| 3321.2.a.i | 16 | 1.a | even | 1 | 1 | trivial | |
| 3321.2.a.j | 16 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3321))\):
|
\( T_{2}^{16} - 20 T_{2}^{14} + 158 T_{2}^{12} - 2 T_{2}^{11} - 629 T_{2}^{10} + 25 T_{2}^{9} + 1329 T_{2}^{8} + \cdots + 3 \)
|
|
\( T_{5}^{16} + T_{5}^{15} - 38 T_{5}^{14} - 24 T_{5}^{13} + 526 T_{5}^{12} + 161 T_{5}^{11} - 3292 T_{5}^{10} + \cdots - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 20 T^{14} + \cdots + 3 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + T^{15} + \cdots - 8 \)
|
| $7$ |
\( T^{16} + 13 T^{15} + \cdots + 1 \)
|
| $11$ |
\( T^{16} - 2 T^{15} + \cdots + 7053 \)
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| $13$ |
\( T^{16} + 5 T^{15} + \cdots + 3 \)
|
| $17$ |
\( T^{16} - T^{15} + \cdots - 8386503 \)
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| $19$ |
\( T^{16} + 19 T^{15} + \cdots + 133137 \)
|
| $23$ |
\( T^{16} + \cdots + 432543603 \)
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| $29$ |
\( T^{16} + \cdots - 609023069 \)
|
| $31$ |
\( T^{16} + 39 T^{15} + \cdots - 1305711 \)
|
| $37$ |
\( T^{16} + \cdots + 168892585947 \)
|
| $41$ |
\( (T + 1)^{16} \)
|
| $43$ |
\( T^{16} + \cdots - 3138869921 \)
|
| $47$ |
\( T^{16} + 5 T^{15} + \cdots - 84757893 \)
|
| $53$ |
\( T^{16} + \cdots - 563452980591 \)
|
| $59$ |
\( T^{16} + \cdots - 2299548164553 \)
|
| $61$ |
\( T^{16} + 21 T^{15} + \cdots - 2822139 \)
|
| $67$ |
\( T^{16} + \cdots - 1326531651448 \)
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| $71$ |
\( T^{16} + \cdots - 4217509809 \)
|
| $73$ |
\( T^{16} + \cdots - 23358486787 \)
|
| $79$ |
\( T^{16} + \cdots - 744927994376 \)
|
| $83$ |
\( T^{16} + \cdots - 42904015439 \)
|
| $89$ |
\( T^{16} + \cdots - 24121243791 \)
|
| $97$ |
\( T^{16} + \cdots - 20834218467 \)
|
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